Non-linear behavior of acoustic waves in solid materials can be related to cracks or other damage in the solid materials and possibly to the state of stress and fluid-saturation. Non-linear behavior in rocks was explored and established by theory and experiments in the late 1980s. An overview is presented in Guyer and Johnson (Non-Linear Mesoscopic Elasticity, Wiley 1999) and Ostrovsky and Johnson (Rivista del Nouvo Cimento, Vol. 24, No. 7, 2001).
The general theory that governs non-linear interaction of acoustic wave in elastic media is well-known as described by Landau & Lifshitz in Theory of Elasticity, 3rd edition, Pergamon Press, Oxford, 1986. There are specific kinematic properties called selection rules that govern non-linear, non-collinear, interactions between two acoustic beams or acoustic plane waves, as disclosed in Jones, G. L. & D. R. Korbett, Interaction of elastic waves in an isotropic solid, J. Acoust. Soc. Am., 35, 5-10 (1963). According to the selection rules, a third plane wave or beam wave resulting from a non-linear mixing of a first acoustic wave and a second acoustic wave is equal to the frequency difference between a frequency f1 of first plane wave and a frequency f2 of the second plane wave. In addition, according to the selection rules, the third plane wave can only be generated for a specific intersecting angle and frequency ratio of the first and second plane waves for any particular Poisson ratio of the medium at the intersecting zone of the first and second plane waves. More complete calculations of the interaction between two non-collinear acoustic plane waves are provided by Korneev, Nihei and Myer. (Nonlinear Interaction of Plane Elastic Waves, Lawrence Berkley National Laboratory, Earth Science Division, June 1998, LBNL 414914).
A basic remote sensing system with non-linear acoustic probes generally consists of two acoustic sources located at two spaced apart positions and an array of acoustic detectors located at a different position from the acoustic sources. The two acoustic sources can generate first and second acoustic waves (e.g., first and second acoustic beam waves) that intersect at various locations in a medium to be investigated. A third acoustic wave (e.g., a third acoustic beam wave) can be generated by a non-linear interaction of the first and second acoustic waves with the medium. The third acoustic wave is then detected at the array of receivers.
Various systems designed for specific applications in a borehole environment have been described by D'Angelo et al. (U.S. Pat. No. 5,521,882), Leggett et al. (U.S. Pat. No. 7,301,852), Khan (U.S. Pat. No. 6,175,536), and Johnson et al. (U.S. Patent Application Publication No. US2010/0265794). These systems provide measurements of complex interference patterns originating from the non-linear interaction of the first and second acoustic waves with the medium.